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Theorem ltexnq 10589
Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexnq (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 10540 . . . 4 <Q ⊆ (Q × Q)
21brel 5614 . . 3 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
3 ordpinq 10557 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
4 elpqn 10539 . . . . . . . . 9 (𝐴Q𝐴 ∈ (N × N))
54adantr 484 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐴 ∈ (N × N))
6 xp1st 7793 . . . . . . . 8 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
75, 6syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐴) ∈ N)
8 elpqn 10539 . . . . . . . . 9 (𝐵Q𝐵 ∈ (N × N))
98adantl 485 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐵 ∈ (N × N))
10 xp2nd 7794 . . . . . . . 8 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
119, 10syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐵) ∈ N)
12 mulclpi 10507 . . . . . . 7 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
137, 11, 12syl2anc 587 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
14 xp1st 7793 . . . . . . . 8 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
159, 14syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐵) ∈ N)
16 xp2nd 7794 . . . . . . . 8 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
175, 16syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐴) ∈ N)
18 mulclpi 10507 . . . . . . 7 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
1915, 17, 18syl2anc 587 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
20 ltexpi 10516 . . . . . 6 ((((1st𝐴) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐵) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴))))
2113, 19, 20syl2anc 587 . . . . 5 ((𝐴Q𝐵Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴))))
22 relxp 5569 . . . . . . . . . . . 12 Rel (N × N)
234ad2antrr 726 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 ∈ (N × N))
24 1st2nd 7810 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2522, 23, 24sylancr 590 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2625oveq1d 7228 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
277adantr 484 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐴) ∈ N)
2817adantr 484 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐴) ∈ N)
29 simpr 488 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝑦N)
30 mulclpi 10507 . . . . . . . . . . . . 13 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3117, 11, 30syl2anc 587 . . . . . . . . . . . 12 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3231adantr 484 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
33 addpipq 10551 . . . . . . . . . . 11 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (𝑦N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
3427, 28, 29, 32, 33syl22anc 839 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
3526, 34eqtrd 2777 . . . . . . . . 9 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
36 oveq2 7221 . . . . . . . . . . . 12 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴))))
37 distrpi 10512 . . . . . . . . . . . . 13 ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦))
38 fvex 6730 . . . . . . . . . . . . . . 15 (2nd𝐴) ∈ V
39 fvex 6730 . . . . . . . . . . . . . . 15 (1st𝐴) ∈ V
40 fvex 6730 . . . . . . . . . . . . . . 15 (2nd𝐵) ∈ V
41 mulcompi 10510 . . . . . . . . . . . . . . 15 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42 mulasspi 10511 . . . . . . . . . . . . . . 15 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
4338, 39, 40, 41, 42caov12 7436 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) = ((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
44 mulcompi 10510 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N 𝑦) = (𝑦 ·N (2nd𝐴))
4543, 44oveq12i 7225 . . . . . . . . . . . . 13 (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦)) = (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴)))
4637, 45eqtr2i 2766 . . . . . . . . . . . 12 (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))) = ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦))
47 mulasspi 10511 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵)))
48 mulcompi 10510 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
4948oveq2i 7224 . . . . . . . . . . . . 13 ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵))) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5047, 49eqtri 2765 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5136, 46, 503eqtr4g 2803 . . . . . . . . . . 11 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)))
52 mulasspi 10511 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
5352eqcomi 2746 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))
5453a1i 11 . . . . . . . . . . 11 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)))
5551, 54opeq12d 4792 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩ = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩)
5655eqeq2d 2748 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩ ↔ (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
5735, 56syl5ibcom 248 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
58 fveq2 6717 . . . . . . . . 9 ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ → ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
59 adderpq 10570 . . . . . . . . . . 11 (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
60 nqerid 10547 . . . . . . . . . . . . 13 (𝐴Q → ([Q]‘𝐴) = 𝐴)
6160ad2antrr 726 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐴) = 𝐴)
6261oveq1d 7228 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
6359, 62eqtr3id 2792 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
64 mulclpi 10507 . . . . . . . . . . . . . . . 16 (((2nd𝐴) ∈ N ∧ (2nd𝐴) ∈ N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6517, 17, 64syl2anc 587 . . . . . . . . . . . . . . 15 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6665adantr 484 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6715adantr 484 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐵) ∈ N)
6811adantr 484 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐵) ∈ N)
69 mulcanenq 10574 . . . . . . . . . . . . . 14 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩)
7066, 67, 68, 69syl3anc 1373 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩)
718ad2antlr 727 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 ∈ (N × N))
72 1st2nd 7810 . . . . . . . . . . . . . 14 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7322, 71, 72sylancr 590 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7470, 73breqtrrd 5081 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵)
75 mulclpi 10507 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (1st𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
7666, 67, 75syl2anc 587 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
77 mulclpi 10507 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (2nd𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
7866, 68, 77syl2anc 587 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
7976, 78opelxpd 5589 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N))
80 nqereq 10549 . . . . . . . . . . . . 13 ((⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵)))
8179, 71, 80syl2anc 587 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → (⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵)))
8274, 81mpbid 235 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵))
83 nqerid 10547 . . . . . . . . . . . 12 (𝐵Q → ([Q]‘𝐵) = 𝐵)
8483ad2antlr 727 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐵) = 𝐵)
8582, 84eqtrd 2777 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = 𝐵)
8663, 85eqeq12d 2753 . . . . . . . . 9 (((𝐴Q𝐵Q) ∧ 𝑦N) → (([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
8758, 86syl5ib 247 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
8857, 87syld 47 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
89 fvex 6730 . . . . . . . 8 ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) ∈ V
90 oveq2 7221 . . . . . . . . 9 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
9190eqeq1d 2739 . . . . . . . 8 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
9289, 91spcev 3521 . . . . . . 7 ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
9388, 92syl6 35 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
9493rexlimdva 3203 . . . . 5 ((𝐴Q𝐵Q) → (∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
9521, 94sylbid 243 . . . 4 ((𝐴Q𝐵Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
963, 95sylbid 243 . . 3 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
972, 96mpcom 38 . 2 (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
98 eleq1 2825 . . . . . . 7 ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q𝐵Q))
9998biimparc 483 . . . . . 6 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
100 addnqf 10562 . . . . . . . 8 +Q :(Q × Q)⟶Q
101100fdmi 6557 . . . . . . 7 dom +Q = (Q × Q)
102 0nnq 10538 . . . . . . 7 ¬ ∅ ∈ Q
103101, 102ndmovrcl 7394 . . . . . 6 ((𝐴 +Q 𝑥) ∈ Q → (𝐴Q𝑥Q))
104 ltaddnq 10588 . . . . . 6 ((𝐴Q𝑥Q) → 𝐴 <Q (𝐴 +Q 𝑥))
10599, 103, 1043syl 18 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
106 simpr 488 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
107105, 106breqtrd 5079 . . . 4 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
108107ex 416 . . 3 (𝐵Q → ((𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
109108exlimdv 1941 . 2 (𝐵Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
11097, 109impbid2 229 1 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wrex 3062  cop 4547   class class class wbr 5053   × cxp 5549  Rel wrel 5556  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  Ncnpi 10458   +N cpli 10459   ·N cmi 10460   <N clti 10461   +pQ cplpq 10462   ~Q ceq 10465  Qcnq 10466  [Q]cerq 10468   +Q cplq 10469   <Q cltq 10472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ni 10486  df-pli 10487  df-mi 10488  df-lti 10489  df-plpq 10522  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-plq 10528  df-mq 10529  df-1nq 10530  df-ltnq 10532
This theorem is referenced by:  ltbtwnnq  10592  prnmadd  10611  ltexprlem4  10653  ltexprlem7  10656  prlem936  10661
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